Semi-log canonical pairs
It was discovered by Kollár and Shepherd-Barron (1988) that in order to compactify the moduli theory of higher dimensional varieties we need stable pairs. That is, pairs (X, Δ) with semi-log canonical singularities and ample log canonical class KX + Δ. The aim of this chapter is to study these semi-log canonical singularities.
In general X is neither normal nor irreducible. Such varieties can be studied either using semi-log resolutions or by focusing on their normalization. Both of these approaches have difficulties.
A stable curve C has ordinary nodes, and we can encode C by giving a triple (, , τ) where is the ...